Representation Theoretical Methods in the Theory of Special Function

拟议的研究涉及计算机探索和信息传输连接数学的几个领域，特别是表示论，理论特殊功能和组合学。 这些联系提供了一个强大的发现工具，因为在一个领域中非常明显的结果和机制往往会转化为其他领域中非常重要和意想不到的事实。这种类型的活动也非常适合培训年轻的研究人员，并为他们提供机会，以发现在我们的计算机时代可以进行一些研究的方式。组合解释将数学信息，无论是代数的，分析的，逻辑的还是其他的，转化为视觉信息。在这种设置下，即使是背景有限的学生也可以体验发现的乐趣。 由于这些原因，PI计划继续通过研究生课程和研讨会将所有当前的研究工作带入课堂的做法。 事实上，PI的学生和合作者在先前的NSF支持下进行的大部分工作都是通过这种课堂活动完成的。代数组合学的子领域，也就是在这个奖项下进行的研究的重点，是由PI的代表性理论方法创建的（1988年）由麦克唐纳关于他现在著名的对称多项式基础。 1990年PI和Mark Haiman的联合工作导致了麦克唐纳多项式空间“对角谐波”的Frobenius特征的简化公式。 PI和Haiman的早期计算机探索产生的数据揭示了对角谐波与“停车函数”（计算机科学家创造的一种丰富多彩的组合结构）的惊人密切联系。这些发现导致了Haglund等人的洗牌猜想的制定，该猜想给出了对角谐波的Frobenius特征的一个漂亮的停车函数的明确公式。大约在2000年，Mark Haiman通过代数几何工具证明了与PI共同制定的原始结构。 这引起了注意的AlgesteurGeometers这一领域的调查。 这种关注导致了一个真正令人惊讶的丰富领域的各种新工具和开放的问题。 最近PI和他的合作者，利用这些新发现成功地制定了一个无限的家庭“洗牌猜想”连接整个李代数的对称函数算子新的“停车功能”一样的对象。 在该奖项的支持下，PI计划使用在该领域二十年的努力中开发的大量工具，与他目前的博士生艾米丽莱文，Yeonkyung Kim和Marino Romero合作解决这些问题。PI和他现在的学生取得的初步成果非常有希望。事实上，最近Leven已经成功地解决了这些新拓扑的一个无限子族。从历史上看，难题一直是基础数学发现的源泉。我们的调查领域在这方面也不应例外。

The proposed research involves computer explorations and the transfer of information connecting several areas of mathematics, most particularly Representation Theory, the Theory Special Functions and Combinatorics. These connections provide a powerful vehicle of discovery, since results and mechanisms which may be quite obvious in one area often translate into highly nontrivial and unexpected facts in one of the other areas. This type of activity is also highly suitable for training young researchers and providing them with the opportunity to discover the manner in which some research can be carried in our Computer Age. Combinatorial interpretations translate mathematical information, be it algebraic, analytical, logical or otherwise, into visual information. Under this setting, even students with limited background can be brought to experience the joy of discovery. For these reasons the PI, plans to continue the practice of bringing all current research efforts right into the classroom through graduate courses and seminars. In fact most of the work of the students and collaborators of the PI carried out under prior NSF support resulted from such classroom activities. The subfield of Algebraic Combinatorics, that is the focus of the research carried out under this award was created by the PI's representation theoretical approach to the (1988) conjectures by Macdonald concerning his now-famous symmetric polynomial basis. The 1990's joint work of the PI and Mark Haiman led to a conjectured formula for the Frobenius characteristic of the space ``Diagonal Harmonics'' in terms of the Macdonald polynomials. Early computer explorations by the PI and Haiman yielded data which revealed a surprisingly intimate connection of Diagonal Harmonics with ``Parking Functions'' (a colorful combinatorial structure created by computer scientists.) These discoveries led to the formulation by Haglund et al. of the Shuffle Conjecture which gives the Frobenius Characteristic of Diagonal Harmonics a beautiful explicit formula in terms of Parking Functions. Around the year 2000, Mark Haiman proved the original conjectures formulated jointly with the PI by Algebraic Geometrical tools. This brought the attention of the Algebraic Geometers to this field of investigation. This attention resulted in a truly surprising enrichment of the field with a variety of new tools and open problems. Very recently the PI and his collaborators, using these new findings succeeded in formulating an infinite family of ``Shuffle Conjectures'' connecting a whole Lie Algebra of Symmetric Function Operators to new ``Parking Function'' like objects. Under the support of this award the PI plans to use the vast collection of tools developed in two decades of efforts in this area to work on these problems in collaboration with his present PhD students Emily Leven, Yeonkyung Kim and Marino Romero. Preliminary results obtained by the PI and his present students have been very promising. In fact, recently Leven has succeeded in solving an infinite subfamily of these new conjectures. Historically, difficult problems have been the source of fundamental mathematical discoveries. Our particular area of investigation should be no exception in this respect.